TITLE: Surjective morphism from $X$ to itself is finite
QUESTION [8 upvotes]: Let $X$ be a projective variety. Why is any surjective morphism from $X$ to itself finite?

REPLY [10 votes]: Let $f$ be your surjective endomorphism; it is generically finite, say of degree $d$. Then  $f_*:H^*(X,\mathbb{Q})\rightarrow H^*(X,\mathbb{Q})$ is surjective, because $\ f_*f^*=d.\mathrm{Id}\ $ (use $\mathbb{Q}_\ell$ instead of  $\mathbb{Q}$ in characteristic $p$). Therefore $f_*$ is bijective. But then $f$ cannot contract any positive dimensional subvariety $Z$, because we would have $f_*[Z]=0$.